It is difficult to diagnose the problem without seeing it however for fungal growth to occur the environment must be suitable for them. They would need enough moisture  so is it potentially wet there? Mushrooms most likely wouldn't be growing on the iron of the stove, so is there wood they are growing on? Perhaps the wood is decaying and they are a saprophytic type of mushroom that is breaking it down. 

Are the mushrooms are the outside of the wood stove or inside?

Since this is really out of our area of expertise, we would recommend that you contact a home restoration company to asses the safety of the mushrooms and to see why they may be growing- a water leak or rotting wood potentially. 

Emily